Proceedings, Eleventh International Conference on Principles of Knowledge Representation and Reasoning (2008)
Formalising Temporal Constraints on Part-Whole Relations
1
2
Alessandro Artale1 , Nicola Guarino2 , and C. Maria Keet1
Faculty of Computer Science, Free University of Bozen-Bolzano, Italy, {artale, keet}@inf.unibz.it
National Research Council, Institute for Cognitive Sciences and Technologies, Trento, Italy, guarino@loa-cnr.it
since we normally assume that each human has necessarily a
specific brain, while not necessarily a specific heart (thanks
to heart transplantation). In addition, if a human is also a
boxer, he must necessarily have exactly his own two hands.
Standard discussions on parthood relationships distinguish
between essential (the brain) and mandatory (the heart)
parts of a whole (the human). In UML modeling, a further case of parthood relationship is introduced: that of socalled immutable parts (Barbier et al. 2003; Guizzardi 2005;
2007). This is indeed the case of the boxer’s hands, which
must exist as long as a human is a boxer (assuming hand
transplantation is forbidden for boxers), but are neither essential nor mandatory for the human’s life.
Abstract
Representing part-whole relations and effectively using
them in domain ontologies and conceptual data models poses multiple challenges. In this paper we face the
issue of imposing temporal constraints on part-whole
relationships, introducing a way to account for essential and immutable parts (and wholes) in addition to the
usual mandatory parts (and wholes). Our approach is
based on i) an explicit temporalization of the part-whole
relation, which allows us to introduce a novel notion
of status for part-whole relationships; ii) an explicit account of the ontological nature of the classes involved
in a part-whole relationships, which distinguishes between rigid and anti-rigid classes. The main novelty
in this paper is to resort to a temporal logic approach to
capture the above mentioned notions. The formalization
proposed here is grounded on the temporal description
logic DLRU S and is based on previous successful efforts to formalize temporal conceptual models.
1
1
Human
1
1
Brain
Heart
1
Introduction
Boxer
The proper account of parthood relations in knowledge representation languages and formal conceptual data models
has received much attention in the literature recently. For
instance, issues such as the transitivity of part-whole relations and the specific behavior of functional parts can be
considered as relatively well clarified nowadays (Varzi 2006;
Vieu 2006; Keet & Artale 2008). However, some subtle issues concerning the various ways two classes may be related by a formula containing modal constraints on parthood
relations are still a matter of discussion in the conceptual
modelling community. Moreover, a proper formalization for
such modal constraints in terms of representation languages
with well-understood computational properties, such as description logics, is still lacking.
Consider, for instance, the UML class diagram shown
in Fig. 1, which constrains the part-whole relations in the
following way: every human has exactly one brain, one
heart and at most two hands, while every boxer (a subclass of human) has exactly two hands. As discussed by
Guizzardi (Guizzardi 2005; 2007), the intended interpretation of these part-whole relationships is typically different,
1
2
0..2
Hand
Figure 1: UML diagram with part-whole relations.
Clearly, cardinality constraints alone are not enough to
distinguish between these interpretations: some kinds of
modal constraints have to be taken into account. A representation language with “full” modal operators would be
the obvious choice, but it would certainly present computational and semantic problems. For this reason, we adopt in
this paper a temporal logic approach, assuming that a temporal modality is enough to capture most practical cases,
and reducing therefore the constraints concerning the socalled “alethic modality”—pure possibility/necessity, independently of time—to those expressible in terms of temporal modality only. In particular, we shall concentrate in this
paper on what we may call the life cycle semantics of partwhole relationships, focusing on the constraints concerning
the temporal existence of both their participants—the part
and the whole—and the part-whole relationship itself. The
explicit recognition given to such distinct temporal behaviors is one of the main contributions of this paper.
Currently, no commonly used conceptual data mod-
c 2008, Association for the Advancement of Artificial
Copyright Intelligence (www.aaai.org). All rights reserved.
673
elling language, such as UML and EER (Extended EntityRelationship), is explicitly tailored to account for these distinctions, although efforts in this direction pointed out useful
modeling patterns and clarified some expressivity requirements (Artale et al. 1996; Barbier et al. 2003; Guizzardi
2005; 2007; Keet 2006; Motschnig-Pitrik & Kaasboll 1999).
In this paper we focus in particular on the formal clarification of the notions of mandatory, essential and immutable
parts when used in conceptual models. The formalization
proposed here is grounded on the temporal description logic
DLRU S and is based on previous efforts to formalize temporal conceptual models. Namely, we rely on a previous
work to define the temporal EER model ERV T that can be
fully captured with DLRU S (Artale et al. 2002; Artale,
Franconi, & Mandreoli 2003; Artale, Parent, & Spaccapietra
2007). The choice of using DLRU S is motivated by its ability to logically reconstruct and extend representational tools
such as object-oriented and conceptual data models, framebased and web ontology languages (Berardi, Calvanese, &
De Giacomo 2005; Calvanese, Lenzerini, & Nardi 1999;
Horrocks, Patel-Schneider, & van Harmelen 2003). In particular, we will show that DLRU S has the expressivity required to formalize the above mentioned distinctions, while
being able, at the same time, to capture two relevant notions
like the rigidity properties of a class (Guarino & Welty 2000;
Welty & Andersen 2005) and the (newly introduced) notion
of status for relations. Concerning the use of automated reasoning services to check quality properties of a conceptual
model we have to remember here that DLRU S is an undecidable language (Artale et al. 2002). A promising direction
is to resort to the a weaker but decidable temporal DL, TDLLite (first results appeared in (Artale et al. 2007)), that temporally extends the computationally simpler DL-Lite with
both temporal roles and concepts. It will be part of our further work to investigate the expressive power of TDL-Lite
to capture temporal conceptual models.
In the remainder of this paper, we first discuss the notions of essential, immutable and mandatory parts exemplified by means of the running example. We then analyse related works on temporal part-whole relations and provide an
overview of the description logic DLRU S . We proceed by
introducing and formalising the notion of status for relations,
and then we show how axioms in DLRU S capture the distinction between rigid and anti-rigid classes. The main section of the paper deals with the constraints on parts, wholes,
and part-whole relations, proving that our formalism easily captures mandatory, essential, and immutable parts (and
wholes). We close with conclusions and future work.
relationship between the class that describes the whole and
the one that describes the part—denoted in the following as
Whole and Part, respectively. Mandatory parts express a
generic dependence relationship in the sense that, although
a part of a certain kind must be always present when the
whole exists, the particular part can be different at different
moments of time (namely the part can be replaced, as in the
human’s heart example). On the other hand, essential parts
express a specific dependence relationship, that is, the whole
must be always associated with the very same part—i.e., the
part must be the same along the entire lifetime of the whole.
If we assume that Whole is rigid (Guarino & Welty 2000;
Welty & Andersen 2005) (that is, if something is a Whole,
then it is a Whole forever), then the distinction between essential and mandatory parts—corresponding to the distinction between specific and generic dependence—is enough to
capture all the useful parthood behaviors.
If Whole is not rigid, however, it is useful to introduce a
further notion, a kind of weaker version of ‘essential part’:
the notion of immutable part. Immutable parts, like essential
parts, are bound to the instances of Whole by a specific dependence relationship. However, this relationship does not
hold necessarily through the whole life of such individuals,
but only as long as they are instances of the class Whole.
Immutable parts are therefore only conditionally essential
for being a Whole, but not essential tout court. For this
reason, immutable parts might also be called conditionally
essential parts. Consider again the class Boxer in Fig. 1,
which we assume to be not rigid (indeed, anti-rigid). Every
boxer must not only have exactly two hands, but he must
have his own hands: if he does not, then he ceases to be a
boxer, while still being a human. In this case, having two
(specific) hands as parts is conditionally essential for being a boxer. If, on the other hand, boxing regulations allow
for substitution of hands (be it by hand transplantation or
protheses), then having some hands is just mandatory for the
boxer. In sum, what we need to account for is another kind
of constraint for the parthood relationship that only holds for
the time the Whole property holds. As we shall see, this will
be formalized by introducing a “guard” condition in the definition for specific dependence which will lead to the notion
of conditionally essential parts (and, in analogy, conditionally essential wholes).
Temporal part-whole relations: Related works
Given that our focus is on temporal modality, let us briefly
discuss related works on temporal part-whole relations. The
most straightforward, yet also limited, way to temporalize
part-whole relations is to turn a part-of predicate from a
binary into a ternary relation, such that we have p part of
w at time t: part of (p, w, t). To the best of our knowledge, almost all current temporalizations of parthood take
this approach (Bittner & Donnelly 2007; Masolo et al. 2003;
Smith et al. 2005)1 , but do not go further to take advantage
of a temporal knowledge representation language. An ex-
Preliminaries on essential, immutable and
mandatory parts
Considering the conceptual model of Fig. 1, we would like
to distinguish between the brain, which we assume to be an
essential part of a human, the heart, which we assume to be
a mandatory part for humans, and the two hands, which we
assume to be immutable parts of every boxer.
The distinction between essential and mandatory parts can
be explained in terms of a specific vs. generic dependence
1
We focus here on temporal parts of objects (i.e., endurants in
DOLCE). Neither parts of events nor parts of 4-dimensional entities (Hawley 2004) are considered here.
674
notes an n-ary relation whose i-th argument (i ≤ n), named
Ui , is of type C. If it is clear from the context, we omit n
and write (Ui : C). The projection expression ∃≶k [Uj ]R is
a generalisation with cardinalities of the projection operator
over argument Uj of relation R; the classical projection is
∃≥1 [Uj ]R.
The model-theoretic semantics of DLRU S assumes a
flow of time T = Tp , <, where Tp is a set of time points
and < a binary precedence relation on Tp , assumed to be isomorphic to Z, <. The language of DLRU S is interpreted
in temporal models over T , which are triples of the form
.
I = T , ΔI , ·I(t) , where ΔI is non-empty set of objects
(the domain of I) and ·I(t) an interpretation function. Since
the domain, ΔI , is time independent, we assume here the
so called constant domain assumption with rigid designator—i.e., an instance is always present in the interpretation
domain and it identifies the same instance at different points
in time. The interpretation function is such that, for every
t ∈ T (a shortcut for t ∈ Tp ), every class C, and every n-ary
relation R, we have C I(t) ⊆ ΔI and RI(t) ⊆ (ΔI )n . The
semantics of class and relation expressions is defined in the
lower part of Fig. 2, where (u, v) = {w ∈ T | u < w < v}.
For classes, the temporal operators 3+ (some time in the
future), ⊕ (at the next moment), and their past counterparts
can be defined via U and S: 3+ C ≡ U C, ⊕ C ≡ ⊥ U C,
etc. The operators 2+ (always in the future) and 2− (always
in the past) are the duals of 3+ (some time in the future)
and 3− (some time in the past), respectively, i.e., 2+ C ≡
¬3+ ¬C and 2− C ≡ ¬3− ¬C, for both classes and relations. The operators 3∗ (at some moment) and its dual 2∗
(at all moments) can be defined for both classes and relations
as 3∗ C ≡ C 3+ C 3− C and 2∗ C ≡ C 2+ C 2− C,
respectively.
A knowledge base is a finite set Σ of DLRU S axioms
of the form C1 C2 and R1 R2 , with R1 and R2 being relations of the same arity. An interpretation I satisfies
C1 C2 (R1 R2 ) if and only if the interpretation of C1
(R1 ) is included in the interpretation of C2 (R2 ) at all time,
I(t)
I(t)
I(t)
I(t)
⊆ C2 (R1 ⊆ R2 ), for all t ∈ T . Thus,
i.e., C1
DLRU S axioms have a global reading. To see examples on
how a DLRU S knowledge base looks like we refer to the
following sections where examples are provided.
ception is (Barbier et al. 2003), where life-time dependencies are represented in UML class diagrams by introducing
an oclUndefined observer function to “assert that all parts
of result do not exist before (@pre)” the creation of the
whole instance w. Barbier and colleagues also tried a representation of immutability, which, however, remained an
open problem due to the lack of a full implementation of
temporal UML. In addition, (Barbier et al. 2003) discussed
nine basic life-cycle cases, obtained by fixing the lifespan
of the Whole and varying the lifespan of the Part. Clearly,
one can consider further cases corresponding to the opposite
perspective, obtained by fixing the part’s lifespan, and assessing its temporal relationships with the whole’s lifespan
(see also Fig. 5). This may sound as a simple inverse, but
we shall see in the following sections that these two views
require distinct constrains.
Regarding the ternary temporal part-whole relation, we
have, for instance, Bittner and Donnelly’s “temporal mereology” (Bittner & Donnelly 2007), which was developed to
deal with “portions of stuff”, i.e., to deal with amounts of
matter, such as gold, and mixtures, such as lemonade, varying in time.
Formalizing temporal aspects by limiting oneself to ad
hoc ternary part-whole relations runs into rather complicated formalizations. On the contrary, well-defined temporal logics—and those applied to temporal conceptual data
modeling in particular—can hide at least some of the details, therefore enhancing understandability and (re)usability
of conceptual models both for the modeler and the domain
expert. This is the reason why we adopt in this paper a
well-studied temporal description logic, DLRU S , which is
already used to model temporal conceptual models (see the
ERV T language in (Artale, Franconi, & Mandreoli 2003;
Artale, Parent, & Spaccapietra 2007)), and we will show
how it enables us to model essential, immutable, and mandatory parts and wholes in a precise and clear way.
The temporal Description Logic DLRU S
The temporal description logic DLRU S (Artale et al. 2002)
combines the propositional temporal logic with Since and
Until operators with the (non-temporal) description logic
DLR (Calvanese & De Giacomo 2003), which serves as
common foundational language for various conceptual data
modeling languages (Calvanese, Lenzerini, & Nardi 1999).
DLRU S can be regarded as an expressive fragment of
the first-order temporal logic L{since, until} (Chomicki &
Toman 1998; Hodgkinson, Wolter, & Zakharyaschev 1999;
Gabbay et al. 2003).
The basic syntactical types of DLRU S are classes and nary relations (n ≥ 2). Starting from a set of atomic classes
(denoted by CN ), a set of atomic relations (denoted by RN ),
and a set of role symbols (denoted by U ), we can define inductively (complex) class and relation expressions (see upper part of Fig. 2), where the binary constructors (, , U, S)
are applied to relations of the same arity, i, j, k, n are natural numbers, i ≤ n, j does not exceed the arity of R, and
all the Boolean constructors are available for both class and
relation expressions. The selection expression Ui /n : C de-
Status Relations and Rigid Classes
The formalization of essential and immutable parts is based
on two preliminary notions that will be introduced in this
section: the original contribution of status relations and the
distinction between rigid and anti-rigid classes.
Status relations
Status relations extend the notion of status classes (Spaccapietra, Parent, & Zimanyi 1998; Etzion, Gal, & Segev
1998) to statuses for relations. Status classes—already
formalized in DLRU S in (Artale, Parent, & Spaccapietra
2007)—constrain the evolution of an instance’s membership
in a class along its lifespan. According to (Spaccapietra, Parent, & Zimanyi 1998; Artale, Parent, & Spaccapietra 2007),
675
C
R
→ | ⊥ | CN | ¬C | C1 C2 | C1 C2 | ∃≶k [Uj ]R |
3+ C | 3− C | 2+ C | 2− C | ⊕ C | C | C1 U C2 | C1 S C2
→ n | RN | ¬R | R1 R2 | R1 R2 | Ui /n : C |
3+ R | 3− R | 2+ R | 2− R | ⊕ R | R | R1 U R2 | R1 S R2
I(t)
⊥I(t)
CN I(t)
(¬C)I(t)
(C1 C2 )I(t)
(C1 C2 )I(t)
(∃≶k [Uj ]R)I(t)
(C1 U C2 )I(t)
(C1 S C2 )I(t)
=
=
⊆
=
=
=
=
=
=
ΔI ;
∅;
I(t) ;
I(t) \ C I(t) ;
I(t)
I(t)
C1 ∩ C2 ;
I(t)
I(t)
C1 ∪ C2 ;
I(t)
{d∈
| {d1 , . . . , dn ∈ RI(t) | dj = d} ≶ k};
I(v)
I(w)
I(t)
{d∈
| ∃v > t.(d ∈ C2 ∧ ∀w ∈ (t, v).d ∈ C1 )};
I(v)
I(w)
{ d ∈ I(t) | ∃v < t.(d ∈ C2 ∧ ∀w ∈ (v, t).d ∈ C1 )};
(n )I(t)
RN I(t)
(¬R)I(t)
(R1 R2 )I(t)
(R1 R2 )I(t)
(Ui /n : C)I(t)
(R1 U R2 )I(t)
(R1 S R2 )I(t)
(3+ R)I(t)
(⊕ R)I(t)
(3− R)I(t)
( R)I(t)
⊆
⊆
=
=
=
=
=
=
=
=
=
=
(ΔI )n ;
(n )I(t) ;
(n )I(t) \ RI(t) ;
I(t)
I(t)
R1 ∩ R 2 ;
I(t)
I(t)
R1 ∪ R 2 ;
{d1 , . . . , dn ∈ (n )I(t) | di ∈ C I(t) };
I(v)
I(w)
{d1 , . . . , dn ∈ (n )I(t) | ∃v > t.(d1 , . . . , dn ∈ R2 ∧ ∀w ∈ (t, v). d1 , . . . , dn ∈ R1 )};
I(v)
I(w)
{ d1 , . . . , dn ∈ (n )I(t) | ∃v < t.(d1 , . . . , dn ∈ R2 ∧ ∀w ∈ (v, t). d1 , . . . , dn ∈ R1 )};
{d1 , . . . , dn ∈ (n )I(t) | ∃v > t. d1 , . . . , dn ∈ RI(v) };
{d1 , . . . , dn ∈ (n )I(t) | d1 , . . . , dn ∈ RI(t+1) };
{d1 , . . . , dn ∈ (n )I(t) | ∃v < t. d1 , . . . , dn ∈ RI(v) };
{d1 , . . . , dn ∈ (n )I(t) | d1 , . . . , dn ∈ RI(t−1) }.
Figure 2: Syntax and semantics of DLRU S .
status modeling includes up to four different statuses: scheduled, active, suspended, disabled, where each one entails
different constraints.
Concerning status relations there are two options: (1) to
derive a relation’s status from the status of the classes participating in the relation, or (2) to explicitly define it on the
relation itself, where the latter, in turn, puts constraints on
the statuses of the classes. Since we are interested in modeling relations as first-class citizens, we choose to have a
means to explicitly model the status of a relation. Therefore, as for classes, we have four different statuses for relations, too—scheduled, active, suspended, disabled—each
illustrated with an example before we proceed to the formal
characterization.
• Scheduled: a relation is scheduled if its instantiation
is known but its membership will only become effective
some time later. Objects in its participating classes must
be either scheduled, too, be active, or suspended. For instance, a pillar for finishing the interior of the Sagrada
Familia in Barcelona is scheduled to become part of that
church, i.e., this part of relation between the pillar and
the church is scheduled.
• Active: the status of a relation is active if the particular relation fully instantiates the type-level relation: the
part is currently part of the whole. For instance, the Mont
Blanc mountain is part of the Alps mountain range, and
the country Republic of Ireland is part of the European
Union. Only active classes can participate into an active
relation.
• Suspended: to capture a temporarily inactive relation.
For example, an instance of a CarEngine is removed from
the instance of a Car it is part of, for purpose of maintenance at the car mechanic. Note that at the moment of
suspension, both the part and the whole must be active,
but can upon suspension of the relation be either active or
become suspended, too, but neither scheduled (see below
constraints on scheduled) nor disabled.
• Disabled: to model expired relations that never again can
be used. For instance, to represent the donor of an organ
who has donated that organ and one wants to keep track
of who donated what to whom: say, the heart p1 of donor
w1 used to be a structural part of w1 but it will never be
again a part of it. The heart, p1 , then may have become
participant in a new part-of relation with a new whole, w2
where w1 = w2 , but the original part-of between p1 and
w1 remains disabled. Observe that participating objects
can be member of the active, suspended or disabled class.
We assume that active relations involve only active classes
and, by default, the name of a relation denotes already its ac-
676
P ROPOSITION 1 (Status Relations: Logical Implications).
Given the set of axioms Σst and an n-ary relation, R U1 :
C1 . . . Un : Cn , the following logical implications hold:
TopR
d
Exists-R
(RACT) Active will possibly evolve into Suspended or Disabled.
Σst |= R 2+ (R Suspended-R Disabled-R)
(RD ISAB 3) Disabled will never become active anymore.
Σst |= Disabled-R 2+ ¬R
(RD ISAB 4) Disabled classes can participate only in disabled relations.
Σst |= Disabled-Ci 3− ∃[Ui ]R ∃[Ui ]Disabled-R
(RD ISAB 5) Disabled relations involve active, suspended,
or disabled classes.
Disabled-R Ui : (Ci Suspended-Ci Disabled-Ci ),
for all i = 1, . . . , n.
(RS CH 3) Scheduled persists until active.
Σst |= Scheduled-R Scheduled-R U R
(RS CH 4) Scheduled cannot evolve directly to Disabled.
Σst |= Scheduled-R ⊕ ¬Disabled-R
(RS CH 5) Scheduled relations do not involve disabled
classes.
Scheduled-R Ui : ¬Dibabled-Ci , for all i = 1, . . . , n.
Disabled-R
.
d
Scheduled-R
R
Suspended-R
Figure 3: Status relations.
tive status—i.e., Active-R ≡ R. Disjointness and ISA constraints among the four status relations are analogous to the
one for status classes and can be represented with an EER
diagram as illustrated in Fig. 3 (double arrows indicate covering and an encircled d means disjointness). In addition to
hierarchical constraints, the following constraints hold (we
present both the model-theoretic semantics and the correspondent DLRU S axioms considering, without loss of generality, binary relations only):
(ACT) Active relations involve only active classes.
I(t)
I(t)
o1 , o2 ∈ RI(t) → o1 ∈ C1 ∧ o2 ∈ C2
R U1 : C1 U2 : C2
Proof. See Appendix.
(RE XISTS) Existence persists until Disabled.
o1 , o2 ∈ Exists-RI(t) → ∀t > t.(o1 , o2 ∈
Exists-RI(t ) ∨ o1 , o2 ∈ Disabled-RI(t ) )
+
Exists-R 2 (Exists-R Disabled-R)
Lifespan and related notions. The lifespan of an object with respect to a class describes the set of temporal
instants where the object can be considered a member of
that class. We can distinguish between the following notions: E XISTENCE S PANC , L IFE S PANC , ACTIVE S PANC ,
B EGINC , B IRTHC , and D EATHC depending on the status
of the class, C, the object is member of. We briefly report
here their definition as presented in (Artale, Parent, & Spaccapietra 2007).
(RD ISAB 1) Disabled persists.
o1 , o2 ∈ Disabled-RI(t) → ∀t > t.o1 , o2 ∈
Disabled-RI(t )
Disabled-R 2+ Disabled-R
(RD ISAB 2) Disabled was Active in the past.
o1 , o2 ∈ Disabled-RI(t) → ∃t < t.o1 , o2 ∈ RI(t )
−
Disabled-R 3 R
E XISTENCE S PANC (o) = {t ∈ T | o ∈ Exists-CI(t) }
L IFE S PANC (o) = {t ∈ T | o ∈ CI(t) ∪ Suspended-CI(t) }
ACTIVE S PANC (o) = {t ∈ T | o ∈ CI(t) }
B EGINC (o) = min(E XISTENCE S PANC (o))
B IRTHC (o) = min(ACTIVE S PANC (o)) ≡ min(L IFE S PANC (o))
D EATHC (o) = max(L IFE S PANC (o))
(RS USP 1) Suspended was Active in the past.
o1 , o2 ∈ Suspended-RI(t) → ∃t < t.o1 , o2 ∈ RI(t )
−
Suspended-R 3 R
(RS USP 2) Suspended involve Active or Suspended Classes.
o1 , o2 ∈ Suspended-RI(t) → oi ∈ Ci I(t) ∨ oi ∈
Suspended-Ci I(t) , i = 1, 2
Suspended-R Ui : (Ci Suspended-Ci ), i = 1, 2
Representing Rigidity
As the running example in Fig. 1 shows, there is a peculiar difference between human and boxer, the latter being a
“role” that a human can play, while a human is not necessarily a boxer and if a human ceases to be a boxer, then he
is still a human. Such differences have been investigated in
detail in (Guarino & Welty 2000), which forms the basis of
the OntoClean methodology (Guarino & Welty 2004). For
the purpose of this paper, we only require the distinctions
between rigid properties vs. non-rigid and anti-rigid properties. We repeat these three definitions here, comment on
them and provide examples in Fig. 4.
(RS CH 1) Scheduled will eventually become Active.
o1 , o2 ∈ Scheduled-RI(t) → ∃t > t.o1 , o2 ∈ RI(t )
+
Scheduled-R 3 R
(RS CH 2) Scheduled can never follow Active.
o1 , o2 ∈ RI(t) → ∀t > t.o1 , o2 ∈ Scheduled-RI(t )
R 2+ ¬Scheduled-R
In the following we denote with Σst the above set of
DLRU S axioms that formalize status relations. Given Σst ,
the following logical implications are relevant to model immutable parts.
Definition 1 (+R). A rigid property φ is a property that
is essential to all its instances, i.e., ∀xφ(x) → φ(x).
677
axioms hold globally, 2∗ means “at all moments”, and 3∗
means “at some moment”).
Definition 2 (-R). A non-rigid property φ is a property that is not essential to some of its instances, i.e.,
∃xφ(x) ∧ ¬φ(x).
(R IGID) Rigid Class
C 2∗ C
Definition 3 (∼R). An anti-rigid property φ is a
property that is not essential to all its instances, i.e.,
∀xφ(x) → ¬φ(x).
Non-sortal
Role
Property
Anti-rigid
Non-rigid
Sortal
Rigid
(A-R IGID) Anti-Rigid Class
C 3∗ ¬C
In addition, we have the constraint that for each anti-rigid
class CA we must have a rigid class CR that subsumes
CA (Guarino & Welty 2000; Guizzardi 2005):
Category
Endurant, Abstract entity
Attribution
Blue, Spherical
Formal role
Recipient
Material role
Student, Food
(A- SUB -R) Each Anti-Rigid Class is a subclass of a Rigid Class
CA CR
Phased sortal
Caterpillar, Chrysalis, Butterfly (for Papilionoidae)
Mixin
Finally, due to the constant domain assumption in
DLRU S , we can capture the notion of existential rigidity
by introducing an exist class, E, collecting, at each point in
time, the objects that actually exist.
Type
Cat, Chair
Quasi-type
Herbivore
(E XIST-R IGID) Existentially Rigid Class
C 2∗ (E → C)
Figure 4: Taxonomy of properties with several examples
(Source: based on (Guarino & Welty 2000)).
Given the above formalization for anti-rigid classes, then
for each object, o, member of an anti-rigid class, CA , we have
that:
Practically, for conceptual data modeling and ontology
development the anti-rigid properties are highly useful to
represent entity types such as boxer and, more generally,
those object types for which object migration may apply.
For example, a human, h1 , may be recorded in a university
database as an instance of PostDoc during some time and be
promoted to Professor at a later time. A common example
for bio-ontologies is the transformation from caterpillar to
butterfly where the same entity changes membership from
one class to another.
Furthermore, as pointed out in (Welty & Andersen 2005),
in the literature different kinds of rigidity have been introduced. Here we also consider the so called existential rigidity that limits rigidity to the actual existence of the instance
(captured by the exist predicate “E(x)”).
ACTIVE S PANCA (o) ⊂ ACTIVE S PANCR (o)
On the other hand, for rigid and existentially rigid classes
the notions of E XISTENCE S PANC (o), L IFE S PANC (o) and
ACTIVE S PANC (o) all denote the full set of time points, T ,
or the set ACTIVE S PANE (o), respectively.
We now have sufficient ingredients to formally characterize the different kinds of dependence between a whole (part)
and its part (whole).
Representing mandatory, immutable, and
essential parts and wholes
The distinction between essential, immutable and mandatory parts (Guizzardi 2007) can be formalized in terms of
two criteria: i) the nature of the dependence relationship between the class that describes the whole and the one that
describes the part (which can be either specific or generic),
and ii) the strength of the membership between the whole
and the class that describes it (which, depending whether
the class is rigid or not, will result in an unconditional or
conditional dependence). We thus distinguish between the
following different cases.
Definition 4 (Existential Rigidity). An existentially
rigid property φ is a property that is essential to all
its instances as long as they exist, i.e., ∀xφ(x) →
(E(x) → φ(x)).
To capture the distinction between immutable and essential parts and wholes, we need to capture the distinction
between rigid and anti-rigid classes. Indeed, while essential parts are properties of a whole that cannot change without destroying the whole, the fact that a part-whole relation
is immutable is conditional to the whole belonging contingently to a particular class in a given interval of time. Thus,
while essential parts are properties of a whole as being member of a rigid class, immutable parts describe properties of a
whole as being member of an anti-rigid class. Furthermore,
since for those instances that belong to the same non-rigid
class at all points in time it is more appropriate to speak of
essential parts, we will formally speak of immutable parts
only in the stricter case of an anti-rigid class.
Starting from Definition 1 and Definition 3, we can enforce a class C to be either rigid or anti-rigid with the following DLRU S axioms, respectively (remember that: DLRU S
1. Generic Dependence – Mandatory Part. The whole must
have a part at each instant of its lifetime. Thus, the presence of the part is mandatory, but it can be replaced over
time (e.g., the human heart example).
2. Unconditional Specific Dependence – Essential Part.
The part is mandatory, but it cannot be replaced without
destroying the whole (e.g., the human brain example).
3. Conditional Specific Dependence – Immutable Part (also
called conditionally essential part). The part is mandatory and cannot be replaced, but only as long as the whole
belongs to the class that describes it (e.g., the boxer’s hand
example).
678
In a symmetric way we can define the notions of mandatory, immutable, and essential wholes. Furthermore, we say
that a part is exclusive if it can be part of at most one whole
(similarly for exclusive wholes).
In the following we provide a formalization using an axiomatization expressed with the temporal description logic
DLRU S of such mandatory, immutable, essential and exclusive parts and wholes.
A. whole's lifetime time
p1
p2
p3
p4
B. part's lifetime
time
w1
w2
w3
w4
Figure 5: Lifelines of immutable parts (p1 . . . p4) w.r.t. the
lifeline of the whole (A)—and vice versa (B).
Mandatory and exclusive parts and wholes
Let PartWhole part : P whole : W be a generic partwhole relation between a whole, W, and a part, P. The following DLRU S axioms give a formalization of mandatory
and exclusive parts and wholes:
ACTIVE S PANP (op ) = T and op is (an active) part of ow
at each instant t ∈ T . Thus, essential parts are global properties of rigid wholes that can be formalized in DLRU S with
the following axioms:
(M AN P) Mandatory Part
W ∃[whole]PartWhole
(M AN W) Mandatory Whole
P ∃[part]PartWhole
(E XL P) Exclusive Part
P ∃≤1 [part]PartWhole
(E XLW) Exclusive Whole
W ∃≤1 [whole]PartWhole
(R IGID) Rigid Whole
W 2∗ W
The human’s heart example can thus be represented by introducing the part-whole relation HumanHeartPW as a subrelation of the generic PartWhole relation:
HumanHeartPW PartWhole
HumanHeartPW part : Heart whole : Human
and then constraining every human to participate at least
once in the HumanHeartPW relation by adding the following (M AN P) axiom:
Human ∃[whole]HumanHeartPW
(E SS W) Essential Whole
P ∃[part]2∗ PartWhole
(E SS P) Essential Part
W ∃[whole]2∗ PartWhole
While essential wholes can be captured as:
(R IGID) Rigid Part
P 2∗ P
By substituting the above rigidity axioms with the existential rigidity axiom we can capture essential parts and
wholes just along the existence lifespan of an object, i.e., it
must be that ACTIVE S PANW (ow ) = ACTIVE S PANP (op ) =
ACTIVE S PANE (ow ), where E is the exist class.
The human’s brain example can be represented by introducing the part-whole relation HumanBrainPW as a subrelation of the generic PartWhole relation:
Given the semantics of DLRU S axioms, a generic heart
must be always associated to a human, i.e., the heart is possibly a different heart at different times but each human must
have a heart along its lifetime. While this example specifies a mandatory part (heart) for a rigid class (human), in the
general case mandatory parts can be specified regardless of
the meta-properties concerning rigidity of the whole.
Finally, if we want to stress the exclusiveness of the heart,
i.e., that a given heart can be part of at most one human, then
we add the (E XL P) axiom:
Heart ∃≤1 [part]HumanHeartPW
Note that to represent mandatory and exclusive parts
(wholes) we do not need to use the temporal operators of
DLRU S .
HumanBrainPW PartWhole
HumanBrainPW part : Brain whole : Human
and then adding both the (R IGID) and (E SS P) axioms:
Human 2∗ Human
Human ∃[whole]2∗ HumanBrainPW
This asserts that the very same brain is a part of the very
same human in all possible time instants. Obviously, exclusiveness axioms can be added to further constrain essential
parts and wholes.
One can also conceive examples of exclusive essential parts or wholes for artifacts, such as that each
DisposableLighter has as an essential part at most
one Firestone, or that each USBStick has exactly one
MemoryCard as essential part.
Essential parts and wholes
Immutable parts and wholes
Essential parts express a specific dependence between the
whole and the correspondent part as opposed to the generic
dependence of a mandatory part. Furthermore, the very
same part (i.e., the specific part) must actively exist during
the entire existence of the whole, unconditionally whether
the whole belongs to different classes during its existence. Given an instance, ow , of the rigid class, W, then,
op is an essential part of ow if ACTIVE S PANW (ow ) =
As we said above, immutable parts are a form of conditionally essential parts—i.e., the part is mandatory and cannot
be replaced, but only as long as the whole belongs to the
class that describes it. Thus, given an anti-rigid class, WA , together with its rigid super-class, WR , and an individual whole,
ow , that is member of WA , with ACTIVE S PANWA (ow ) ⊂
ACTIVE S PANWR (ow ), then, op is an immutable part of ow if
679
ACTIVE S PANWA (ow ) ⊆ ACTIVE S PANP (op ) and op is (an active) part of ow at each instant t ∈ ACTIVE S PANWA (ow ). The
temporal relations p1, . . . , p4 of Fig. 5-A illustrate exactly
all the possible different temporal relationships between the
activespan of the given whole (top line of the figure) and the
activespan of its immutable part.
A similar reading can be associated to Fig. 5-B, where
w1, . . . , w4 now denote all the possible different temporal
relations between a fixed part and its immutable whole. Note
that the temporal relation p2 (w2) can be regarded as the
most general notion of immutable part (whole).
To formalize the notion of immutable parts (wholes) in
DLRU S , we need to resort to the notion of status for both
classes and relations as illustrated previously. In addition,
the whole (part) must belong to an anti-rigid class2 .
Let us show now that DLRU S has enough expressivity to
account for the various cases of immutable parts and wholes
introduced above. To see that, we shall first introduce a set
of useful DLRU S axioms expressing some basic constraints
and then we shall prove that each of the possibilities described by Fig. 5-A (Fig. 5-B) can be expressed by suitable
combination of these axioms. The DLRU S axioms are as
follows:
3. p3 if (M AN P), (S US W), (D IS W), (S CH PW), (S CH P)
hold.
4. p1 if (M AN P), (S US W), (D IS W), (D IS P), (S CH PW),
(S CH P) hold.
Proof. See Appendix.
A similar result can be proved for immutable wholes. In
analogy with the (S US W) axiom, we need to introduce an
axiom that governs the case in which the part is suspended:
(S US P) If part-whole is suspended then the part is suspended.
Suspended-PartWhole part : Suspended-P
T HEOREM 3 (Immutable Wholes). Let PR be a rigid class
(i.e., PR 2∗ PR ), P be an anti-rigid class (i.e., P 3∗ ¬P)
s.t. P PR , and PartWhole part : P whole : W be a
generic part-whole relation satisfying Σst . Then, for each
part, op , of type P there exists an immutable whole, ow , of
type W that is temporally related to op with the relation:
1. w2 if (M AN W), (S US P), (D IS P) hold.
2. w4 if (M AN W), (S US P), (D IS P), (D IS W) hold.
3. w3 if (M AN W), (S US P), (D IS P), (S CH PW), (S CH W)
hold.
4. w1 if (M AN W), (S US P), (D IS P), (D IS W), (S CH PW),
(S CH W) hold.
(S US W) If part-whole is suspended then the whole is suspended.
Suspended-PartWhole whole : Suspended-W
(D IS P) If part-whole is disabled then the part is disabled.
Disabled-PartWhole part : Disabled-P
Proof. Similar to Theorem 2.
Observe that our formalization of immutability allows for
suspension of parts, wholes and their part-whole relations,
thanks to the introduction of status for both classes and relations. Since status is not considered in the literature for
immutability of part-whole relations (Barbier et al. 2003;
Guizzardi 2005; 2007), we can enforce immutability to hold
just for continuously active classes. To achieve this, it is
enough to substitute the (S US P) and (S US W) axioms with
the following axiom that forbids suspension of the partwhole relation:
(D IS W) If part-whole is disabled then the whole is disabled.
Disabled-PartWhole whole : Disabled-W
(S CH PW) The part-whole was scheduled sometime in the
past.
PartWhole 3− Scheduled-PartWhole
(S CH P) If part-whole is scheduled then the part is scheduled.
Scheduled-PartWhole part : Scheduled-P
(C ON PW) Continuous active part-whole relation.
Suspended-PartWhole ⊥
(S CH W) If part-whole is scheduled then the whole is scheduled.
Scheduled-PartWhole whole : Scheduled-W
Using the (C ON PW) axiom, the immutable parts and
wholes are always active while participating in the partwhole relation and cannot be suspended. Furthermore, when
the stricter case is allowed (i.e. either p1 or w1 holds), then
they are either both member of their respective Scheduled
class, or both Active, or both member of their respective
Disabled classes. Hence, a change of status from one of the
two objects implies an instantaneous change of the other in
the same type of status class.
The boxer’s hands example can be represented with the
following axioms—enforcing e.g. case p2. First, we introduce the part-whole relation HumanHandPW as a sub-relation
of the generic PartWhole relation:
HumanHandPW PartWhole
HumanHandPW part : Hand whole : Human
The following Theorem shows the DLRU S axioms
needed to represent the various forms of immutable parts
as illustrated in Fig. 5-A.
T HEOREM 2 (Immutable Parts). Let WR be a rigid class
(i.e., WR 2∗ WR ), W be an anti-rigid class (i.e., W 3∗ ¬W)
s.t. W WR , and PartWhole part : P whole : W be a
generic part-whole relation satisfying Σst . Then, for each
whole, ow , of type W there exists an immutable part, op , of
type P that is temporally related to ow with the relation:
1. p2 if (M AN P), (S US W), (D IS W) hold.
2. p4 if (M AN P), (S US W), (D IS W), (D IS P) hold.
2
To capture immutable parts for the more general case of nonrigid classes, our approach forces the introduction of an anti-rigid
sub-class typing the instances having a limited lifetime for which
we can speak of immutable parts.
then we assert that boxer is anti-rigid and it is a sub-class of
the rigid class human:
680
Boxer 3∗ ¬Boxer
Boxer Human
and finally we add the (M AN P), (S US W), (D IS W) axioms
rephrased in terms of the HuamnHandPW part-whole relation:
Boxer ∃=2 [whole]HumanHandPW
Suspended-HumanHandPW whole : Suspended-Boxer
Disabled-HumanHandPW whole : Disabled-Boxer
The use of the (S US W) axiom instead of (C ON PW) captures the scenario where it is permitted for a boxer to be
temporarily member of the Suspended-Boxer class during
which the part-whole relation is suspended as well (e.g., the
boxer has a broken hand that is healing). To be compatible
with extant literature on immutability, we can enforce a continuously active boxer by replacing the (S US W) axiom with
the (C ON PW) axiom:
Suspended-HumanHandPW ⊥
Thinking of examples of immutable parts for artifacts,
we may consider, e.g., an EcoFarm (anti-rigid class) as a
sub-class of RealEstate (rigid class), where an area of
Farmland is an immutable part of the EcoFarm: if, say, the
ecofarm’s farmland is confiscated, then the residual real estate becomes member of either the Suspended-EcoFarm or
Disabled-EcoFarm class (while continuing to be an active
member of RealEstate).
Finally, observe that exclusiveness and immutability are
orthogonal notions in our modeling framework. The formalization of immutable parts and wholes can be easily extended by adding to the axiomatization of Theorems 2-3 either (E XL P) or (E XLW), depending whether we want to capture exclusive immutable parts or wholes.
Conclusions
In this paper we proposed a formalization of mandatory, immutable, and essential parts and wholes adopting a temporal
logic approach. For each different parthood property we presented its formalization with the aim to clarify their meaning
when used in a conceptual modelling language. The temporal description logic DLRU S has been adopted thanks to its
already established ability to give a logical reconstruction of
(temporal) conceptual modelling languages. In addition to
represent the different interpretations of a part-whole relation, we introduced the novel notion of status for relations
that temporally constrains the evolution of a relation. The
notion of status relations has been showed crucial to capture
immutable parts and has been formalized in DLRU S .
Concerning the computational properties, we are aware
of the undecidability of DLRU S . A promising direction is
to resort to the a weaker but decidable temporal DL, TDLLite (Artale et al. 2007) that temporally extends DL-Lite
with both temporal roles and concepts. We are currently
working on both extending the framework of TDL-Lite with
sub-roles and showing the ability of TDL-Lite to represent
temporal conceptual models.
Acknowledgement
We would like to thank Giancarlo Guizzardi for the extensive discussions we had with him. We also thank the anony-
681
mous reviewers for their insightful comments.
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(RS CH 3) Scheduled persists until active.
Σst |= Scheduled-R Scheduled-R U R
(RS CH 4) Scheduled cannot evolve directly to Disabled.
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(RS CH 5) Scheduled relations do not involve disabled
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Proof. We prove here just (RD ISAB 4). The others cases
are similar to what was already proved in (Artale, Parent,
& Spaccapietra 2007).
I(t)
(RD ISAB 4).
Let oi ∈ Disabled-Ci
and r =
o1 , . . . , oi , . . . , on ∈ RI(t ) for some t < t. Then, by
(RACT), r ∈ (R Suspended-R Disabled-R)I(t) . Since
active relations, by (ACT), can involve just active classes,
then, r ∈ RI(t) . On the other hand, by (RS USP 2), r ∈
Suspended-RI(t) . Thus, r ∈ Disabled-RI(t) .
T HEOREM 2 (Immutable Parts). Let WR be a rigid class
(i.e., WR 2∗ WR ), W be an anti-rigid class (i.e., W 3∗ ¬W)
s.t. W WR , and PartWhole part : P whole : W be a
generic part-whole relation satisfying Σst . Then, for each
whole, ow , of type W there exists an immutable part, op , of
type P that is temporally related to ow with the relation:
1. p2 if (M AN P), (S US W), (D IS W) hold.
2. p4 if (M AN P), (S US W), (D IS W), (D IS P) hold.
3. p3 if (M AN P), (S US W), (D IS W), (S CH PW), (S CH P)
hold.
4. p1 if (M AN P), (S US W), (D IS W), (D IS P), (S CH PW),
(S CH P) hold.
Proof. Let ow ∈ WI(t0 ) with t0 = B IRTHW (ow ), then by
(M AN P), ∃op ∈ PI(t0 ) and op , ow ∈ PartWholeI(t0 ) .
I(t )
Since W WR , then, ow ∈ WR 0 , while since WR is rigid
and W is anti-rigid, then, ACTIVE S PANWR (ow ) = T and
ACTIVE S PANW (ow ) ⊂ T . Thus, ACTIVE S PANW (ow ) ⊂
ACTIVE S PANWR (ow ).
C ASE p2. To prove that p2 of Fig. 5 holds we shall
prove that ACTIVE S PANW (ow ) ⊆ ACTIVE S PANP (op )—
i.e., B IRTHW (ow ) ≥ B IRTHP (op ) and D EATHW (ow ) ≤
D EATHP (op )—and op is an active part of ow at each
instant t ∈ ACTIVE S PANW (ow ).
Now, let t0 <
t < D EATHW (ow ) and ow ∈ WI(t ) .
Then, by
(RACT), either (i) op , ow ∈ PartWholeI(t ) , or (ii)
op , ow ∈ Suspended-PartWholeI(t ) or (iii) op , ow ∈
Disabled-PartWholeI(t ) . The last two cases cannot hap
pen since, by assumption ow ∈ WI(t ) , while: (iii) by
(D IS W), ow ∈ Disabled-WI(t ) ; (ii) by (S US W), ow ∈
Suspended-WI(t ) . Thus, op is an active part of ow and,
Appendix
P ROPOSITION 1 (Status Relations: Logical Implications).
Given the set of axioms Σst and an n-ary relation, R U1 :
C1 . . . Un : Cn , the following logical implications hold:
(RACT) Active will possibly evolve into Suspended or Disabled.
Σst |= R 2+ (R Suspended-R Disabled-R)
(RD ISAB 3) Disabled will never become active anymore.
Σst |= Disabled-R 2+ ¬R
682
since by assumption active relations involve only active
classes, op ∈ PI(t ) , for all t s.t. t0 < t < D EATHW (ow ).
For t1 ≥ D EATHW (ow ) and t2 ≤ B IRTHW (ow ) none of the
axioms constraints the lifespan of op . Thus, D EATHW (ow ) ≤
D EATHP (op ), and B IRTHW (ow ) ≥ B IRTHP (op ).
C ASE p1.
To prove that the case p1 of Fig. 5
holds we should prove that ACTIVE S PANW (ow ) = AC TIVE S PAN P (op )—i.e., B IRTH W (ow ) = B IRTH P (op ) and
D EATHW (ow ) = D EATHP (op )—and op is an active part of
ow at each instant t ∈ ACTIVE S PANW (ow ). As for case
p2, since (S US W) and (D IS W) hold, then, op is an ac
tive part of ow , and op ∈ PI(t ) , for all t s.t. t0 <
t < D EATHW (ow ). Now, let t1 = D EATHW (ow ), then, by
(RD ISAB 4), op , ow ∈ Disabled-PartWholeI(t1 ) , and,
by (D IS P), op ∈ Disabled-PI(t1 ) . Thus, D EATHW (ow ) =
D EATHP (op ). Now, by absurd, let’s assume that op ∈
P I(t2 ) , with t2 < t0 . By (S CH PW), there is a t <
t0 s.t. op , ow ∈ Scheduled-PartWholeI(t ) and, by
(S CH P), op ∈ Scheduled-PI(t ) . Then, t2 = t . Also,
t2 < t since, by (S CH 2), an active class cannot evolve
into its scheduled status. Finally, t0 > t2 > t since, by
(RS CH 3) op , ow ∈ Scheduled-PartWholeI(t2 ) and, by
(S CH P), op ∈ Scheduled-PI(t2 ) . Thus, B IRTHW (ow ) =
B IRTHP (op ).
Cases p4, p3 can be easily obtained from the above cases.
683